Question

Suppose we wish to prove that for all integers ≥ 3, 2n + 1 < 2n by using mathematical induction.

a. Show that the statement is true for = 3.
b. What is the inductive hypothesis?
c. What must be proved?
d. Complete the proof.

Answer 1

To prove 2n + 1 < 2n for all integers n ≥ 3 using mathematical induction, we show it is true for n = 3 and then prove it holds for the next integer.

Step-by-step explanation:

To prove the statement 2n + 1 < 2n for all integers n ≥ 3 using mathematical induction:

a. We show that it is true for n = 3: 2(3) + 1 < 2(3), which simplifies to 7 < 6.

b. The inductive hypothesis is that the statement is true for some arbitrary integer k ≥ 3, i.e., 2k + 1 < 2k.

c. The statement that needs to be proved is that the inequality holds true for the next integer k + 1, i.e., 2(k + 1) + 1 < 2(k + 1).

d. To complete the proof, we assume the inductive hypothesis is true for k and then manipulate the inequality to show that it holds for k + 1. By distributing and simplifying, we can establish that 2k + 3 < 2k + 2, which confirms the statement for the next integer.